# Optimal Control of a Continuous Stirred-Tank Reactor

Optimal Control of a Continuous Stirred-Tank Reactor

Aris and Amundson [1–3] originally analyzed the design of a continuous stirred-tank reactor (CSTR) with a first-order, irreversible exothermic reaction. The reactor is controlled by varying the flow of a cooling fluid through a coil inside the reactor. The state equations for the CSTR are given by

dT

dt

25T

T+2

1

dC

dt

25T

T+2

where and are the deviations from steady-state concentration and temperature, respectively, and is the control action.

C

T

u

1

This Demonstration finds the optimal control, which minimizes the performance index given by , where the normalized control action is and is a weighting parameter. Assume a final time of the control [4, 5]. You can set the initial conditions (i.e. both and ) and choose to apply upper and lower bounds on .

J=∫[T+C+Ru]dt

t

f

0

2

2

2

u=u-1

1

R=0.1

t=0.78

f

T(t=0)

C(t=0)

u

We plot the component of the state vector ( and shown in red and blue, respectively), the normalized control , and the costates. We also compute the value of the performance index .

T

C

u

J

For the case where is unbounded, if you select the initial conditions and , then , in close agreement with the result found in [4] and [5].

u

T(t=0)=0.05

C(t=0)=0

J=0.02660

The costates and are plotted in magenta and cyan. The costates verify and (yellow dot).

λ(t)

1

λ(t)

2

λ(t)=0

1

f

λ(t)=0

2

f